Fractal Wallpapers - Symmetry II

More about the Wallpaper Groups..

Cutting a background element

Cutting a background element out of a screenshot

Producing Wallpapers

You can produce your own endless background images with the Wallpaper Applet: Choose an IFS figure and the wallpaper mode. Form the IFS by moving a single fixed point or move the whole IFS (hold the CTRL-key while moving a circle) to another position of the wallpaper machine. Both influences the result. Do a screenshot and cut a minimal pattern for your background image like the example shown above.

Some hints:
With an IFS, which produces a figure of high density, the tesselation of the Euclidean plane looks nicer, if the tesselation elements don't overlap each other. By moving the fixed points to shorter distances You can make the IFS smaller. It is also possible to adjust the tesselation factor (in a range of 0.1 to 4.0) in order to give more room (> 1.0) to the base figure. IF the basic figure is thin, overlapping could be a good idea to produce exciting tesselations, look for the IFS titled "Drawing" in the applet.

Names of the Wallpaper Groups

A wallpaper ornament fills the whole Euclidean plane with a basic pattern. 17 different modes are possible and therefor used, described by cryptic names like CMM, P31M and P6M. Watch the results to understand how the transformations work. The following table explains the symbols used to describe these transformations.

 Symbol    Meaning


The letter P means that there is a primitve cell.


The letter C is used to say that there is a rhombic cell with at least one diagonal as a mirror axis. The cell could be integrated into a double sized rectangle cell, positioned as a centered cell.


The letter M tells us that there is a mirror transformation.


The letter G announces a glide reflection. A glide reflection is a mirror reflection followed by a vector move.

2, 3, 4, 6

The numbers tell us about centers of a n-ordered rotation symmetry. Only these numbers are possible. An example: The number 3 says, that a rotation of 120° will leave the pattern unchanged (3×120° = 360°).


The number 1 in P31M signalizes that there are no mirror or glide reflection axis perpendicularly to the borders of the primitive cell. In P3M1 there aren't such axis which cut the borders with an angle of 60°.


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© 2007 Ulrich Schwebinghaus